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I assume in my question that all resistors follow Ohm's law.

As I understand Ohm's law can be explained such that if we connect a resistor to a power source that is not time dependent then we have the relationship $I=V/R$, the current will be proprtional to the voltage.

We also know that when we connect many resistors together they will still follow Ohm's law individually and we have the known formulas for series and paralell resistors.

But why do we know that when we connect resistors together in either series or paralell they still follow Ohms'law individually, and the voltage drop over them will not vary in time, and the current over them will not vary in time?

Is this something that is known because it has been shown through experiments? Or is there a theoretical explanation for the fact that when we connect resistors together the voltage drop over each of them will not vary in time, and they will individually follow Ohm's law?

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    $\begingroup$ Your question is unclear. A resistor doesn't know anything about anything around it except for what appears as its two terminals so why would it behave any differently? $\endgroup$
    – DKNguyen
    Commented Sep 3, 2021 at 20:17
  • $\begingroup$ In short, what we as Voltage drop is due to the fact that some work needs to be done to move charges. That work is independent of time until and unless heat comes into action. $\endgroup$
    – Nikola Alfredi
    Commented Sep 3, 2021 at 20:54
  • $\begingroup$ @DKNguyen Yes, but what was confusing me is that when we connect the resistors together they may have some sort of interaction so that we get some sort of time dependence.(I get that this will not happen, but I do not know why.) $\endgroup$
    – user394334
    Commented Sep 3, 2021 at 23:00
  • $\begingroup$ @user394334 Those are called transmission line effects. It's basically when transient currents flow back AND forth producing voltages and such as the different parts of the circuit "probe" to see what's out there before settling into a steady state equilibrium. They always happen but happen very fast and do not last a long time before equilibrium is reached. We only concern ourselves with them when they have significant effect on our circuit (i.e. faster circuits or very long lines where the times involved are on par with the times our circuit signals operates on). $\endgroup$
    – DKNguyen
    Commented Sep 3, 2021 at 23:15
  • $\begingroup$ Analog:y when you pour water into a long pipe you have no idea that the other end is plugged. you only know that the other end is plugged when the water reaches the end and starts backing up and splashes you. $\endgroup$
    – DKNguyen
    Commented Sep 3, 2021 at 23:19

3 Answers 3

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Because ideal resistors are linear elements, when you connect lots of linear elements the collection is also linear.

Further, ideal resistors have no "history" or ability to store energy or information. They behave identically at all times given an applied voltage.

Using the first property we can find that a circuit with only ideal voltage sources and ideal resistors will only have a single steady-state solution.

Given the second property, the circuit approaches that solution immediately, with no ability oscillate or have other behaviors.

A real circuit will have some amount of inductance and capacitance, so that steady-state is not reached instantaneously, but it'll be pretty darn fast.

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  • $\begingroup$ Thank you. Is there a way to show that the current and voltages must all be constant over each resistor? Can one start with the possibility that the current and voltage over each resistor is time dependent, and then solve the equation and then show that both the voltage and current is constant in time? If so, what is this equation called? How do you show that there is only one solution to this? $\endgroup$
    – user394334
    Commented Sep 3, 2021 at 22:28
  • $\begingroup$ How would one for instance show that if two resistors connected in parllell will have only one solution? You don't need to solve the equation, but which equation would you use? $\endgroup$
    – user394334
    Commented Sep 3, 2021 at 23:07
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Your question goes the wrong way. Resistance of an object is defined by $R=V/I$ if you alpine them in a row, the current I will be the same in every one of them- do you understand why?- and then you can measure the voltage over any of them and find out its resistance.

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I assume in my question that all resistors follow Ohm's law.

You can, of course, assume that for the purpose of the question as long as you realize that it is not necessarily true that all resistors follow Ohm's law.

As I understand Ohm's law can be explained such that if we connect a resistor to a power source that is not time dependent then we have the relationship $I=V/R$, the current will be proprtional to the voltage.

It is not necessary for the voltage source to be not time dependent (to be constant). You just need to specify what $I$ is (instantaneous, peak, rms, etc..)

We also know that when we connect many resistors together they will still follow Ohm's law individually and we have the known formulas for series and paralell resistors.

Yes, again assuming the resistance of the resistors is constant (not dependent on voltage, temperature, etc..)

But why do we know that when we connect resistors together in either series or paralell they still follow Ohms'law individually...

As long as their resistances are constant, there is no reason to believe otherwise.

...and the voltage drop over them will not vary in time, and the current over them will not vary in time?

As I already indicated, Ohms law is not restricted to constant power sources, so voltages and currents can vary over time (just not the resistance, if they are ideal), and still be related by Ohm's law.

Is this something that is known because it has been shown through experiments?

Good question. As a matter of fact, Ohm's law is not really a fundamental law, per se. It is based on observations (experiments).

Or is there a theoretical explanation for the fact that when we connect resistors together the voltage drop over each of them will not vary in time, and they will individually follow Ohm's law?

Again, there is no reason why the voltage and current cannot vary in time, as long the the relationship between the two for a purely resistive (no inductance or capacitance) circuit follows Ohm's law. Ohm's law is not limited to a constant voltage source.

Hope this helps.

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